Another idea to further explore the profound connection between spacetime geometry and quantum entanglement is inspired by the optimization problem in network flow theory. [1]}, [2]}, [3]} developed the optimization theory of flows on manifolds, endowed with the name of “bit threads", and proposed the concept of bit threads can equivalently formulate the RT formula. Although bit threads are visually intuitive and usually interpreted implicitly as the bell pairs distilled from the boundary quantum systems, they are mostly used merely as a mathematical tool to study different aspects of holographic principles (for the recent developments of bit threads see e.g. [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}). Among these developments, it is worth noting that in [5]} bit threads are related with the holographic entanglement distillation tensor networks, and in [4]}, the component flow fluxes in the locking bit thread configurations are shown to explain the partial entanglement entropies of the boundary quantum systems. In this paper For a brief pre-release version containing the core of this work, see [24]}., by studying the connection between bit threads and SS duality, we propose a natural and novel physical property of bit threads, dubbed “thread/state correspondence”, that is, in a so-called locking bit thread configuration [6]}, [5]}, [4]}, [28]}, each bit thread is in a quantum superposition state of two orthogonal states. Our thread/state rules explicitly accentuate for the first time the implied meaning of the name “bit threads”, namely, that these threads, which are used mathematically to recover the RT formula, can be physically assigned a meaning closely related to the concept of “bits”. Moreover, since the fluxes of bit threads are directly related to the geometric quantities of a holographic spacetime, while this newly proposed thread-state property can cleverly characterize quantum entanglement as one will see, it can be expected to be a very useful advance in the study of the relationship between quantum entanglement and spacetime geometry.
